Relativistic Treatment of Frequency Shifts for Spin 1/2 Particles in Electromagnetic Fields.

We present a relativistic generalization to the problem of calculating frequency shifts for spin-1/2 particles undergoing Larmor precession in electromagnetic fields. Such effects are important in the interpretation of the results from experiments employing Larmor precession techniques. For the case in which these particles are moving in a highly-uniform magnetic field with small non-uniformities and a non-zero electric field, magnetic fields oriented in the transverse direction relative to the primary magnetic field direction can induce frequency shifts in the particles' Larmor spin precession frequencies. In a situation which employs both electric and magnetic fields, there are, in general, two possible sources of transverse magnetic fields in the particle rest frame: gradients in the magnetic field, leading to off-primary-axis transverse field components, and relativistic transformations of electric fields. Considerations of such are then of paramount importance to the interpretation of results from a precision measurement of a Larmor spin precession frequency.